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Tsallis entropy

In physics, the Tsallis entropy is a generalization of the standard Boltzmann-Gibbs entropy. It was an extension put forward by Constantino Tsallis in 1988. It is defined as

$S_q(p) = {1 \over q - 1} \left( 1 - \int p^q(x)\, dx \right),$

or in the discrete case

$S_q(p) = {1 \over q - 1} \left( 1 - \sum_x p^q(x) \right).$

In this case, p denotes the probability distribution of interest, and q is a real parameter. In the limit as q → 1, the normal Boltzmann-Gibbs entropy is recovered.

The parameter q is a measure of the non-extensitivity of the system of interest. There are continuous and discrete versions of this entropic measure.

Various relationships

The discrete Tsallis entropy satisfies

$S_q = - \left [ D_q \sum_i p_i^x \right ]_{x=1}$

where Dq is the q-derivative.

Non-extensivity

Given two independent systems A and B, for which the joint probability density satisfies

$p(A, B) = p(A) p(B),\,$

the Tsallis entropy of this system satisfies

$S_q(A,B) = S_q(A) + S_q(B) + (1-q)S_q(A) S_q(B).\,$

From this result, it is evident that the parameter q is a measure of the departure from extensivity. In the limit when q = 1,

$S(A,B) = S(A) + S(B),\,$

which is what is expected for an extensive system.